Aims. The multi-wavelength emission from a newly identified population of ‘extreme-TeV’ blazars, with Compton peak frequencies around 1 TeV, is difficult to interpret with standard one-zone emission models. Large values of the minimum electron Lorentz factor and quite low magnetisation values seem to be required.Methods. We propose a scenario where protons and electrons are co-accelerated on internal or recollimation shocks inside the relativistic jet. In this situation, energy is transferred from the protons to the electrons in the shock transition layer, leading naturally to a high minimum Lorentz factor for the latter. A low magnetisation favours the acceleration of particles in relativistic shocks.Results. The shock co-acceleration scenario provides additional constraints on the set of parameters of a standard one-zone lepto-hadronic emission model, reducing its degeneracy. Values of the magnetic field strength of a few mG and minimum electron Lorentz factors of 103 to 104, required to provide a satisfactory description of the observed spectral energy distributions of extreme blazars, result here from first principles. While acceleration on a single standing shock is sufficient to reproduce the emission of most of the extreme-TeV sources we have examined, re-acceleration on a second shock appears needed for those objects with the hardest γ-ray spectra. Emission from the accelerated proton population, with the same number density as the electrons but in a lower range of Lorentz factors, is strongly suppressed. Satisfactory self-consistent representations were found for the most prominent representatives of this new blazar class.